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| """Most of these tests come from the examples in Bronstein's book.""" | |
| from sympy.core.numbers import (I, Rational, oo) | |
| from sympy.core.symbol import symbols | |
| from sympy.polys.polytools import Poly | |
| from sympy.integrals.risch import (DifferentialExtension, | |
| NonElementaryIntegralException) | |
| from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer, | |
| normal_denom, special_denom, bound_degree, spde, solve_poly_rde, | |
| no_cancel_equal, cancel_primitive, cancel_exp, rischDE) | |
| from sympy.testing.pytest import raises | |
| from sympy.abc import x, t, z, n | |
| t0, t1, t2, k = symbols('t:3 k') | |
| def test_order_at(): | |
| a = Poly(t**4, t) | |
| b = Poly((t**2 + 1)**3*t, t) | |
| c = Poly((t**2 + 1)**6*t, t) | |
| d = Poly((t**2 + 1)**10*t**10, t) | |
| e = Poly((t**2 + 1)**100*t**37, t) | |
| p1 = Poly(t, t) | |
| p2 = Poly(1 + t**2, t) | |
| assert order_at(a, p1, t) == 4 | |
| assert order_at(b, p1, t) == 1 | |
| assert order_at(c, p1, t) == 1 | |
| assert order_at(d, p1, t) == 10 | |
| assert order_at(e, p1, t) == 37 | |
| assert order_at(a, p2, t) == 0 | |
| assert order_at(b, p2, t) == 3 | |
| assert order_at(c, p2, t) == 6 | |
| assert order_at(d, p1, t) == 10 | |
| assert order_at(e, p2, t) == 100 | |
| assert order_at(Poly(0, t), Poly(t, t), t) is oo | |
| assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \ | |
| order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1 | |
| assert order_at_oo(Poly(0, t), Poly(1, t), t) is oo | |
| def test_weak_normalizer(): | |
| a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t) | |
| d = Poly(t**4 - 3*t**2 + 2, t) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]}) | |
| r = weak_normalizer(a, d, DE, z) | |
| assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t, domain='ZZ[x]'), | |
| (Poly((1 + x)*t**2 + x*t, t, domain='ZZ[x]'), | |
| Poly(t + 1, t, domain='ZZ[x]'))) | |
| assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1]) | |
| r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z) | |
| assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t))) | |
| assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1]) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]}) | |
| r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z) | |
| assert r == (Poly(t, t), (Poly(0, t), Poly(1, t))) | |
| assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1]) | |
| def test_normal_denom(): | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) | |
| raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x), | |
| Poly(1, x), Poly(x, x), DE)) | |
| fa, fd = Poly(t**2 + 1, t), Poly(1, t) | |
| ga, gd = Poly(1, t), Poly(t**2, t) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]}) | |
| assert normal_denom(fa, fd, ga, gd, DE) == \ | |
| (Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t), | |
| Poly(1, t)), Poly(t, t)) | |
| def test_special_denom(): | |
| # TODO: add more tests here | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]}) | |
| assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t), | |
| Poly(t, t), DE) == \ | |
| (Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t)) | |
| # assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1 | |
| # issue 3940 | |
| # Note, this isn't a very good test, because the denominator is just 1, | |
| # but at least it tests the exp cancellation case | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0), | |
| Poly(I*k*t1, t1)]}) | |
| DE.decrement_level() | |
| assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0), | |
| Poly(1, t0), DE) == \ | |
| (Poly(1, t0, domain='ZZ'), Poly(I*k, t0, domain='ZZ_I[k,x]'), | |
| Poly(t0, t0, domain='ZZ'), Poly(1, t0, domain='ZZ')) | |
| assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t), | |
| Poly(t, t), DE, case='tan') == \ | |
| (Poly(1, t, t0, domain='ZZ'), Poly(t**2, t0, t, domain='ZZ[x]'), | |
| Poly(t, t, t0, domain='ZZ'), Poly(1, t0, domain='ZZ')) | |
| raises(ValueError, lambda: special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t), | |
| Poly(t, t), DE, case='unrecognized_case')) | |
| def test_bound_degree_fail(): | |
| # Primitive | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), | |
| Poly(t0/x**2, t0), Poly(1/x, t)]}) | |
| assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t), | |
| Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t, | |
| t), DE) == 3 | |
| def test_bound_degree(): | |
| # Base | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) | |
| assert bound_degree(Poly(1, x), Poly(-2*x, x), Poly(1, x), DE) == 0 | |
| # Primitive (see above test_bound_degree_fail) | |
| # TODO: Add test for when the degree bound becomes larger after limited_integrate | |
| # TODO: Add test for db == da - 1 case | |
| # Exp | |
| # TODO: Add tests | |
| # TODO: Add test for when the degree becomes larger after parametric_log_deriv() | |
| # Nonlinear | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]}) | |
| assert bound_degree(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), DE) == 0 | |
| def test_spde(): | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]}) | |
| raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE)) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]}) | |
| assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t), | |
| Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \ | |
| (Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t, domain='ZZ(x)')) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]}) | |
| assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t), | |
| Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \ | |
| (Poly(0, t), Poly(0, t), 0, Poly(0, t), | |
| Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t, domain='ZZ(x,t0)')) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) | |
| assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 + | |
| 3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \ | |
| (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), 2, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x)) | |
| assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 + | |
| 3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \ | |
| (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), -2 + n, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x)) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]}) | |
| raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE)) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) | |
| assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == \ | |
| (Poly(0, x, domain='ZZ'), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x, domain='QQ')) | |
| assert spde(Poly(x**2 - x, x), Poly(x**2 - 5*x + 3, x), Poly(x**7 - x**6 - 2*x**4 + 3*x**3 - x**2, x), 5, DE) == \ | |
| (Poly(1, x, domain='QQ'), Poly(x + 1, x, domain='QQ'), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x, domain='QQ')) | |
| def test_solve_poly_rde_no_cancel(): | |
| # deg(b) large | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]}) | |
| assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t), | |
| oo, DE) == Poly(t + x, t) | |
| # deg(b) small | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) | |
| assert solve_poly_rde(Poly(0, x), Poly(x/2 - Rational(1, 4), x), oo, DE) == \ | |
| Poly(x**2/4 - x/4, x) | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]}) | |
| assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \ | |
| Poly(t + 1, t, x) | |
| # deg(b) == deg(D) - 1 | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]}) | |
| assert no_cancel_equal(Poly(1 - t, t), | |
| Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \ | |
| (Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t)) | |
| def test_solve_poly_rde_cancel(): | |
| # exp | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]}) | |
| assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \ | |
| Poly(1, t) | |
| assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \ | |
| Poly(t, t) | |
| # TODO: Add more exp tests, including tests that require is_deriv_in_field() | |
| # primitive | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]}) | |
| # If the DecrementLevel context manager is working correctly, this shouldn't | |
| # cause any problems with the further tests. | |
| raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE)) | |
| assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \ | |
| Poly(t, t) | |
| assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \ | |
| Poly(t**2, t) | |
| # TODO: Add more primitive tests, including tests that require is_deriv_in_field() | |
| def test_rischDE(): | |
| # TODO: Add more tests for rischDE, including ones from the text | |
| DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]}) | |
| DE.decrement_level() | |
| assert rischDE(Poly(-2*x, x), Poly(1, x), Poly(1 - 2*x - 2*x**2, x), | |
| Poly(1, x), DE) == \ | |
| (Poly(x + 1, x), Poly(1, x)) | |